Approximation of supremum of max-stable stationary processes & Pickands constants. (English) Zbl 1462.60039
In this paper explicit formulas and sufficient conditions for the positivity of the Pickands constant \(\mathcal H\) corresponding to the spectral process \(Z\) attached to a stochastically continuous stationary max-stable process \(X(t), t \in\mathbb R\), with Fréchet marginals \(\Phi_\alpha, \alpha>0\), are obtained. Since only stationary max-stable processes are considered here, \(Z\) is a Brown-Resnick stationary process and \(\mathcal H\) is called generalized Pickands constant.
The paper is organized in four sections and an Appendix. After the necessary notions introduced in the first section, in the second section the main results are presented in two main theorems followed by some remarks and examples in the Gaussian case and stationary max-stable Lévy-Brown-Resnick processes. Since Slepian inequality is essential in the theory of extremes and sample path properties of Gaussian and related processes, in the third section of the paper the Slepian inequality for Brown-Resnick max-stable stationary processes \(X_1\) and \(X_2\) is stated and a comparison criterium for the corresponding Picands constants \({\mathcal H}_1\) and \({\mathcal H}_2\) is obtained. Also, for a given Brown-Resnick stationary process \(Z\) the Piterbarg constants are defined and for max-stable processes the finiteness of Piterbarg constants for more general cases is shown and the growth of supremum is analyzed. The proofs of the theorems presented in the previous sections are given in the fourth section. The paper ends with an Appendix presenting the tilt-shift formula for the special case of Brown-Resnick max-stable process with log-normal \(Z\).
The paper is organized in four sections and an Appendix. After the necessary notions introduced in the first section, in the second section the main results are presented in two main theorems followed by some remarks and examples in the Gaussian case and stationary max-stable Lévy-Brown-Resnick processes. Since Slepian inequality is essential in the theory of extremes and sample path properties of Gaussian and related processes, in the third section of the paper the Slepian inequality for Brown-Resnick max-stable stationary processes \(X_1\) and \(X_2\) is stated and a comparison criterium for the corresponding Picands constants \({\mathcal H}_1\) and \({\mathcal H}_2\) is obtained. Also, for a given Brown-Resnick stationary process \(Z\) the Piterbarg constants are defined and for max-stable processes the finiteness of Piterbarg constants for more general cases is shown and the growth of supremum is analyzed. The proofs of the theorems presented in the previous sections are given in the fourth section. The paper ends with an Appendix presenting the tilt-shift formula for the special case of Brown-Resnick max-stable process with log-normal \(Z\).
Reviewer: Ilie Valuşescu (Bucureşti)
MSC:
| 60G10 | Stationary stochastic processes |
| 60G15 | Gaussian processes |
| 60G70 | Extreme value theory; extremal stochastic processes |
Keywords:
max-stable process; spectral tail process; Gaussian processes with stationary increments; Lévy processes; Pickands constants; Piterbarg constants; Slepian inequality; growth of supremumReferences:
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